Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

112 kenneth mandersSuppose the statement to be shown set out in a diagram. Logically, thisinvolves considering an instance; as we hereafter suppose done, in order toavoid quantifiers. Let the statement have the form (C 1 &... &C n ) → D. Weassume that the Cs andDs are what we call diagram conditions: either exactconditions that would be asserted in the discursive text, or explicit co-exactconditions, that could be indicated in or read off from a diagram. The classof diagram conditions is not closed under negation: there is no conditionequivalent to non-circularity, for example.To proceed by reductio, we must consider C 1 &... &C n &¬D. Itremainstoconvert ¬D into a diagram condition. Typically, D asserts an equality, or isstraightforwardly interderivable with one; thus, its negation can be convertedby trichotomy into a disjunction of two strict inequalities (x is greater than y).With luck, as in I.6 (but not I.25), symmetry considerations allow this to bereduced to one. A strict inequality, in turn, may be reduced to a proper-partrelationship in the diagram by using an auxiliary equality:x is greater than y ⇔ forsomeproperpartz of x, z = y.In principle, one could hope to deal with claims of the form, for any mand n,(C 1 &... &C n ) → (D 1 ∨ ... ∨ D m ).When denied as hypothesis for reductio and set out in a diagram, these givethe formC 1 &... &C n &¬D 1 &... &¬D m .In general, though, each of the ¬D j reduces to a disjunction of diagramconditions, and for increasing m putting this as a disjunction of conjunctionsgives an exponential explosion of interacting case distinctions. It is thereforeunlikely that we would encounter, in a single proposition established byreductio, m greater than one.I.7 shows a special case, where in effect m = 0; the statement is thenequivalent to ¬(C 1 &... &C n ). Such assertions are infrequent; perhaps becauserecognizably negative claims are frowned upon, perhaps because negationsof diagram conditions often cannot be converted into diagram conditions,and hence the resulting proposition could only be applied in further reductioarguments. The hypothesis for reductio is then C 1 &... &C n , which is immediatelyof the form to put forward of a diagram. III.13, thattwocirclesdonot have two distinct tangents, and Proclus’ corollary from I.17, that there isonly one perpendicular on one side of a line at a given point (313), are also ofthis type.